Bootstrap Methods

Non-parametric confidence intervals and hypothesis testing for prediction market edge estimation. 11 bootstrap variants + jackknife bias correction.

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What is this? Monte Carlo simulation assumes you know the data-generating distribution. Bootstrap methods don’t - they resample your actual data to estimate confidence intervals, standard errors, and p-values. Use bootstrap when you want to know if your edge is statistically real without making distributional assumptions.

Bootstrap Methods

Horizon provides 11 bootstrap variants and jackknife bias estimation, all implemented in Rust for performance. These are non-parametric alternatives to Monte Carlo that work directly from observed data. The bootstrap is especially valuable in prediction markets where returns are non-normal and sample sizes are small.

IID Bootstrap

bootstrap_mean(), bootstrap_sharpe(), bootstrap_var(), bootstrap_edge() - classic resampling with replacement.

Time-Series Bootstrap

block_bootstrap(), circular_block_bootstrap(), stationary_bootstrap() - preserve autocorrelation structure.

Hypothesis Testing

bootstrap_hypothesis_test() - permutation test to compare two strategies without distributional assumptions.

Jackknife

jackknife_bias() - leave-one-out bias and variance estimation with pseudovalues.

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IID Bootstrap Functions

hz.bootstrap_mean

Confidence interval for the mean via IID resampling.

import horizon as hz

returns = [0.02, -0.01, 0.03, 0.01, -0.005, 0.015, 0.025, -0.008]
r = hz.bootstrap_mean(returns, n_resamples=10000, confidence=0.95, seed=42)
print(f"Mean: {r.point_estimate:.4f}")
print(f"95% CI: [{r.ci_lower:.4f}, {r.ci_upper:.4f}]")
print(f"Std error: {r.std_error:.4f}")

hz.bootstrap_sharpe

Confidence interval for the Sharpe ratio. Particularly useful because the Sharpe ratio’s sampling distribution is highly non-normal for small samples.

r = hz.bootstrap_sharpe(returns, n_resamples=10000, confidence=0.95, seed=42)
print(f"Sharpe: {r.point_estimate:.4f}")
print(f"95% CI: [{r.ci_lower:.4f}, {r.ci_upper:.4f}]")
# If CI includes zero, your Sharpe isn't significantly different from zero

hz.bootstrap_var

Confidence interval for Value-at-Risk.

r = hz.bootstrap_var(returns, var_level=0.05, n_resamples=10000, seed=42)
print(f"5% VaR: {r.point_estimate:.4f}")
print(f"95% CI: [{r.ci_lower:.4f}, {r.ci_upper:.4f}]")

hz.bootstrap_correlation

Confidence interval for Pearson correlation (paired resampling).

x = [0.5, 0.6, 0.7, 0.55, 0.65]
y = [0.48, 0.62, 0.71, 0.52, 0.68]
r = hz.bootstrap_correlation(x, y, n_resamples=10000, seed=42)
print(f"Correlation: {r.point_estimate:.4f}")
print(f"95% CI: [{r.ci_lower:.4f}, {r.ci_upper:.4f}]")

hz.bootstrap_edge

Confidence interval for prediction edge mean(outcome - prediction). This is the core bootstrap for prediction market edge estimation.

predictions = [0.55, 0.60, 0.45, 0.70, 0.50]
outcomes =    [1.0,  1.0,  0.0,  1.0,  0.0]
r = hz.bootstrap_edge(predictions, outcomes, n_resamples=10000, seed=42)
print(f"Edge: {r.point_estimate:.4f}")
print(f"95% CI: [{r.ci_lower:.4f}, {r.ci_upper:.4f}]")
if r.ci_lower > 0:
    print("Statistically significant positive edge!")

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Time-Series Bootstrap

For autocorrelated data (e.g., cumulative returns, price series), IID resampling destroys the dependence structure. These variants preserve it.

hz.block_bootstrap

Moving block bootstrap (Kunsch 1989). Resamples contiguous blocks of data.

prices = [0.50, 0.51, 0.52, 0.51, 0.53, 0.54, 0.52, 0.55]
r = hz.block_bootstrap(prices, block_size=3, n_resamples=5000, seed=42)
print(f"Mean: {r.point_estimate:.4f}, CI: [{r.ci_lower:.4f}, {r.ci_upper:.4f}]")

Parameter	Type	Description
data	list[float]	Time series data
block_size	int	Size of each contiguous block
n_resamples	int	Number of bootstrap resamples (default 10000)
confidence	float	Confidence level (default 0.95)
seed	int	RNG seed (default 42)

hz.circular_block_bootstrap

Politis-Romano circular variant. Wraps the data circularly so blocks can span the boundary. Reduces bias from edge effects.

r = hz.circular_block_bootstrap(prices, block_size=3, n_resamples=5000, seed=42)

hz.stationary_bootstrap

Politis-Romano stationary bootstrap. Block lengths are geometric random variables, making the resampled series stationary.

r = hz.stationary_bootstrap(prices, expected_block_size=5.0, n_resamples=5000, seed=42)

Parameter	Type	Description
expected_block_size	float	Expected block length (geometric distribution)

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Hypothesis Testing

hz.bootstrap_hypothesis_test

Two-sample permutation test. Tests whether two samples come from distributions with the same mean.

strategy_a = [0.02, 0.01, 0.03, -0.01, 0.015]
strategy_b = [0.005, -0.01, 0.01, -0.005, 0.002]

r = hz.bootstrap_hypothesis_test(strategy_a, strategy_b, n_resamples=10000, seed=42)
print(f"Test statistic: {r.test_statistic:.4f}")
print(f"P-value: {r.p_value:.4f}")
print(f"Reject H0: {r.reject}")  # True if p < 0.05

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Prediction Intervals

hz.bootstrap_prediction_interval

Build prediction intervals from residual resampling. Useful for forecasting.

residuals = [-0.05, -0.02, 0.01, 0.03, -0.01, 0.04, -0.03]
forecast = 0.65
r = hz.bootstrap_prediction_interval(residuals, forecast, n_resamples=10000, seed=42)
print(f"Forecast: {r.point_estimate:.4f}")
print(f"95% PI: [{r.ci_lower:.4f}, {r.ci_upper:.4f}]")

---

Jackknife

hz.jackknife_bias

Leave-one-out jackknife for bias and variance estimation.

data = [0.02, 0.01, 0.03, -0.01, 0.015, 0.025]
r = hz.jackknife_bias(data, statistic="mean")
print(f"Estimate: {r.estimate:.4f}")
print(f"Bias: {r.bias:.6f}")
print(f"Std error: {r.std_error:.4f}")
print(f"Pseudovalues: {len(r.pseudovalues)}")

Parameter	Type	Description
data	list[float]	Input data
statistic	str	One of "mean", "sharpe", "var" (default "mean")

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Result Types

BootstrapResult

Field	Type	Description
point_estimate	float	Original sample statistic
ci_lower	float	Lower confidence bound
ci_upper	float	Upper confidence bound
std_error	float	Bootstrap standard error
n_resamples	int	Number of resamples used
bias	float	Bootstrap bias estimate

BootstrapHypothesisResult

Field	Type	Description
test_statistic	float	Observed difference in means
p_value	float	Two-sided p-value
ci_lower	float	Lower CI of permutation distribution
ci_upper	float	Upper CI of permutation distribution
reject	bool	True if p-value < 0.05

JackknifeResult

Field	Type	Description
estimate	float	Full-sample statistic
bias	float	Jackknife bias estimate
std_error	float	Jackknife standard error
pseudovalues	list[float]	Leave-one-out pseudovalues

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Pipeline Integration

hz.bootstrap_edge_tracker

Tracks bootstrap CI for prediction edge in a live pipeline. Accumulates (prediction, outcome) pairs per market and computes CIs when the window fills.

hz.run(
    pipeline=[
        my_model,
        my_quoter,
        hz.bootstrap_edge_tracker(window=100, n_resamples=5000, confidence=0.95),
    ],
    ...
)
# Injects into ctx.params: "bootstrap_edge" = {edge, ci_lower, ci_upper, is_significant}

hz.bootstrap_strategy_test

Compares two strategies’ returns using a permutation test.

hz.run(
    pipeline=[
        hz.bootstrap_strategy_test("returns_a", "returns_b"),
    ],
    ...
)
# Reads "returns_a" and "returns_b" from ctx.params
# Injects: "strategy_test" = {test_statistic, p_value, reject}